OpenRaiser/Intern-Atlas huggingface dataset:
Consensus conditions of continuous-time multi-agent systems with time-delays and measurement noises
Venue: arXiv cs.SY 2018
Year: 2018

This work is concerned with stochastic consensus conditions of multi-agent systems with both time-delays and measurement noises. For the case of additive noises, we develop some necessary conditions and sufficient conditions for stochastic weak consensus by estimating the differential resolvent function for delay equations. By the martingale convergence theorem, we obtain necessary conditions and sufficient conditions for stochastic strong consensus. For the case of multiplicative noises, we consider two kinds of time-delays, appeared in the measurement term and the noise term, respectively. We first show that stochastic weak consensus with the exponential convergence rate implies stochastic strong consensus. Then by constructing degenerate Lyapunov functional, we find the sufficient consensus conditions and show that stochastic consensus can be achieved by carefully choosing the control gain according to the noise intensities and the time-delay in the measurement term.
Cluster-Seeking James-Stein Estimators
Venue: arXiv cs.IT 2018
Year: 2018

This paper considers the problem of estimating a high-dimensional vector of parameters $\boldsymbolθ \in \mathbb{R}^n$ from a noisy observation. The noise vector is i.i.d. Gaussian with known variance. For a squared-error loss function, the James-Stein (JS) estimator is known to dominate the simple maximum-likelihood (ML) estimator when the dimension $n$ exceeds two. The JS-estimator shrinks the observed vector towards the origin, and the risk reduction over the ML-estimator is greatest for $\boldsymbolθ$ that lie close to the origin. JS-estimators can be generalized to shrink the data towards any target subspace. Such estimators also dominate the ML-estimator, but the risk reduction is significant only when $\boldsymbolθ$ lies close to the subspace. This leads to the question: in the absence of prior information about $\boldsymbolθ$, how do we design estimators that give significant risk reduction over the ML-estimator for a wide range of $\boldsymbolθ$? In this paper, we propose shrinkage estimators that attempt to infer the structure of $\boldsymbolθ$ from the observed data in order to construct a good attracting subspace. In particular, the components of the observed vector are separated into clusters, and the elements in each cluster shrunk towards a common attractor. The number of clusters and the attractor for each cluster are determined from the observed vector. We provide concentration results for the squared-error loss and convergence results for the risk of the proposed estimators. The results show that the estimators give significant risk reduction over the ML-estimator for a wide range of $\boldsymbolθ$, particularly for large $n$. Simulation results are provided to support the theoretical claims.
Average Distance in a General Class of Scale-Free Networks with Underlying Geometry
Venue: arXiv cs.DM 2018
Year: 2018

In Chung-Lu random graphs, a classic model for real-world networks, each vertex is equipped with a weight drawn from a power-law distribution, and two vertices form an edge independently with probability proportional to the product of their weights. Chung-Lu graphs have average distance O(log log n) and thus reproduce the small-world phenomenon, a key property of real-world networks. Modern, more realistic variants of this model also equip each vertex with a random position in a specific underlying geometry. The edge probability of two vertices then depends, say, inversely polynomial on their distance. In this paper we study a generic augmented version of Chung-Lu random graphs. We analyze a model where the edge probability of two vertices can depend arbitrarily on their positions, as long as the marginal probability of forming an edge (for two vertices with fixed weights, one fixed position, and one random position) is as in Chung-Lu random graphs. The resulting class contains Chung-Lu random graphs, hyperbolic random graphs, and geometric inhomogeneous random graphs as special cases. Our main result is that every random graph model in this general class has the same average distance as Chung-Lu random graphs, up to a factor 1+o(1). This shows in particular that specific choices, such as the underlying geometry being Euclidean or the dependence on the distance being inversely polynomial, do not significantly influence the average distance. The proof also yields that our model has a giant component and polylogarithmic diameter with high probability.
Recovering Structured Probability Matrices
Venue: arXiv cs.LG 2018
Year: 2018

We consider the problem of accurately recovering a matrix B of size M by M , which represents a probability distribution over M2 outcomes, given access to an observed matrix of "counts" generated by taking independent samples from the distribution B. How can structural properties of the underlying matrix B be leveraged to yield computationally efficient and information theoretically optimal reconstruction algorithms? When can accurate reconstruction be accomplished in the sparse data regime? This basic problem lies at the core of a number of questions that are currently being considered by different communities, including building recommendation systems and collaborative filtering in the sparse data regime, community detection in sparse random graphs, learning structured models such as topic models or hidden Markov models, and the efforts from the natural language processing community to compute "word embeddings". Our results apply to the setting where B has a low rank structure. For this setting, we propose an efficient algorithm that accurately recovers the underlying M by M matrix using Theta(M) samples. This result easily translates to Theta(M) sample algorithms for learning topic models and learning hidden Markov Models. These linear sample complexities are optimal, up to constant factors, in an extremely strong sense: even testing basic properties of the underlying matrix (such as whether it has rank 1 or 2) requires Omega(M) samples. We provide an even stronger lower bound where distinguishing whether a sequence of observations were drawn from the uniform distribution over M observations versus being generated by an HMM with two hidden states requires Omega(M) observations. This precludes sublinear-sample hypothesis tests for basic properties, such as identity or uniformity, as well as sublinear sample estimators for quantities such as the entropy rate of HMMs.
Constrained Generalized Delaunay Graphs Are Plane Spanners
Venue: arXiv cs.CG 2018
Year: 2018

We look at generalized Delaunay graphs in the constrained setting by introducing line segments which the edges of the graph are not allowed to cross. Given an arbitrary convex shape $C$, a constrained Delaunay graph is constructed by adding an edge between two vertices $p$ and $q$ if and only if there exists a homothet of $C$ with $p$ and $q$ on its boundary that does not contain any other vertices visible to $p$ and $q$. We show that, regardless of the convex shape $C$ used to construct the constrained Delaunay graph, there exists a constant $t$ (that depends on $C$) such that it is a plane $t$-spanner of the visibility graph. Furthermore, we reduce the upper bound on the spanning ratio for the special case where the empty convex shape is an arbitrary rectangle to $\sqrt{2} \cdot \left( 2 l/s + 1 \right)$, where $l$ and $s$ are the length of the long and short side of the rectangle.
Modeling cumulative biological phenomena with Suppes-Bayes Causal Networks
Venue: arXiv cs.AI 2018
Year: 2018

Several diseases related to cell proliferation are characterized by the accumulation of somatic DNA changes, with respect to wildtype conditions. Cancer and HIV are two common examples of such diseases, where the mutational load in the cancerous/viral population increases over time. In these cases, selective pressures are often observed along with competition, cooperation and parasitism among distinct cellular clones. Recently, we presented a mathematical framework to model these phenomena, based on a combination of Bayesian inference and Suppes' theory of probabilistic causation, depicted in graphical structures dubbed Suppes-Bayes Causal Networks (SBCNs). SBCNs are generative probabilistic graphical models that recapitulate the potential ordering of accumulation of such DNA changes during the progression of the disease. Such models can be inferred from data by exploiting likelihood-based model-selection strategies with regularization. In this paper we discuss the theoretical foundations of our approach and we investigate in depth the influence on the model-selection task of: (i) the poset based on Suppes' theory and (ii) different regularization strategies. Furthermore, we provide an example of application of our framework to HIV genetic data highlighting the valuable insights provided by the inferred.
Balanced Allocation: Patience is not a Virtue
Venue: arXiv cs.DS 2018
Year: 2018

Load balancing is a well-studied problem, with balls-in-bins being the primary framework. The greedy algorithm $\mathsf{Greedy}[d]$ of Azar et al. places each ball by probing $d > 1$ random bins and placing the ball in the least loaded of them. With high probability, the maximum load under $\mathsf{Greedy}[d]$ is exponentially lower than the result when balls are placed uniformly randomly. Vöcking showed that a slightly asymmetric variant, $\mathsf{Left}[d]$, provides a further significant improvement. However, this improvement comes at an additional computational cost of imposing structure on the bins. Here, we present a fully decentralized and easy-to-implement algorithm called $\mathsf{FirstDiff}[d]$ that combines the simplicity of $\mathsf{Greedy}[d]$ and the improved balance of $\mathsf{Left}[d]$. The key idea in $\mathsf{FirstDiff}[d]$ is to probe until a different bin size from the first observation is located, then place the ball. Although the number of probes could be quite large for some of the balls, we show that $\mathsf{FirstDiff}[d]$ requires only at most $d$ probes on average per ball (in both the standard and the heavily-loaded settings). Thus the number of probes is no greater than either that of $\mathsf{Greedy}[d]$ or $\mathsf{Left}[d]$. More importantly, we show that $\mathsf{FirstDiff}[d]$ closely matches the improved maximum load ensured by $\mathsf{Left}[d]$ in both the standard and heavily-loaded settings. We further provide a tight lower bound on the maximum load up to $O(\log \log \log n)$ terms. We additionally give experimental data that $\mathsf{FirstDiff}[d]$ is indeed as good as $\mathsf{Left}[d]$, if not better, in practice.
On the Distributed Complexity of Large-Scale Graph Computations
Venue: arXiv cs.DC 2018
Year: 2018

Motivated by the increasing need to understand the distributed algorithmic foundations of large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where $k \geq 2$ machines jointly perform computations on graphs with $n$ nodes (typically, $n \gg k$). The input graph is assumed to be initially randomly partitioned among the $k$ machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication {\em rounds} of the computation. Our main contribution is the {\em General Lower Bound Theorem}, a theorem that can be used to show non-trivial lower bounds on the round complexity of distributed large-scale data computations. The General Lower Bound Theorem is established via an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic and this theorem can be used in a "cookbook" fashion to show distributed lower bounds in the context of several problems, including non-graph problems. We present two applications by showing (almost) tight lower bounds for the round complexity of two fundamental graph problems, namely {\em PageRank computation} and {\em triangle enumeration}. Our approach, as demonstrated in the case of PageRank, can yield tight lower bounds for problems (including, and especially, under a stochastic partition of the input) where communication complexity techniques are not obvious. Our approach, as demonstrated in the case of triangle enumeration, can yield stronger round lower bounds as well as message-round tradeoffs compared to approaches that use communication complexity techniques.
Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques
Venue: arXiv cs.IT 2018
Year: 2018

We consider a graphical approach to index coding. While cycles have been shown to provide coding gain, only disjoint cycles and cliques (a specific type of overlapping cycles) have been exploited in existing literature. In this paper, we define a more general form of overlapping cycles, called the interlinked-cycle (IC) structure, that generalizes cycles and cliques. We propose a scheme, called the interlinked-cycle-cover (ICC) scheme, that leverages IC structures in digraphs to construct scalar linear index codes. We characterize a class of infinitely many digraphs where our proposed scheme is optimal over all linear and non-linear index codes. Consequently, for this class of digraphs, we indirectly prove that scalar linear index codes are optimal. Furthermore, we show that the ICC scheme can outperform all existing graph-based schemes (including partial-clique-cover and fractional-local-chromatic number schemes), and a random-coding scheme (namely, composite coding) for certain graphs.
Network Unfolding Map by Edge Dynamics Modeling
Venue: arXiv cs.AI 2018
Year: 2018

The emergence of collective dynamics in neural networks is a mechanism of the animal and human brain for information processing. In this paper, we develop a computational technique using distributed processing elements in a complex network, which are called particles, to solve semi-supervised learning problems. Three actions govern the particles' dynamics: generation, walking, and absorption. Labeled vertices generate new particles that compete against rival particles for edge domination. Active particles randomly walk in the network until they are absorbed by either a rival vertex or an edge currently dominated by rival particles. The result from the model evolution consists of sets of edges arranged by the label dominance. Each set tends to form a connected subnetwork to represent a data class. Although the intrinsic dynamics of the model is a stochastic one, we prove there exists a deterministic version with largely reduced computational complexity; specifically, with linear growth. Furthermore, the edge domination process corresponds to an unfolding map in such way that edges "stretch" and "shrink" according to the vertex-edge dynamics. Consequently, the unfolding effect summarizes the relevant relationships between vertices and the uncovered data classes. The proposed model captures important details of connectivity patterns over the vertex-edge dynamics evolution, in contrast to previous approaches which focused on only vertex or only edge dynamics. Computer simulations reveal that the new model can identify nonlinear features in both real and artificial data, including boundaries between distinct classes and overlapping structures of data.
Compositional game theory
Venue: arXiv cs.GT 2018
Year: 2018

We introduce open games as a compositional foundation of economic game theory. A compositional approach potentially allows methods of game theory and theoretical computer science to be applied to large-scale economic models for which standard economic tools are not practical. An open game represents a game played relative to an arbitrary environment and to this end we introduce the concept of coutility, which is the utility generated by an open game and returned to its environment. Open games are the morphisms of a symmetric monoidal category and can therefore be composed by categorical composition into sequential move games and by monoidal products into simultaneous move games. Open games can be represented by string diagrams which provide an intuitive but formal visualisation of the information flows. We show that a variety of games can be faithfully represented as open games in the sense of having the same Nash equilibria and off-equilibrium best responses.
Tensor Methods and Recommender Systems
Venue: arXiv cs.LG 2018
Year: 2018

A substantial progress in development of new and efficient tensor factorization techniques has led to an extensive research of their applicability in recommender systems field. Tensor-based recommender models push the boundaries of traditional collaborative filtering techniques by taking into account a multifaceted nature of real environments, which allows to produce more accurate, situational (e.g. context-aware, criteria-driven) recommendations. Despite the promising results, tensor-based methods are poorly covered in existing recommender systems surveys. This survey aims to complement previous works and provide a comprehensive overview on the subject. To the best of our knowledge, this is the first attempt to consolidate studies from various application domains in an easily readable, digestible format, which helps to get a notion of the current state of the field. We also provide a high level discussion of the future perspectives and directions for further improvement of tensor-based recommendation systems.
A Generic Inverted Index Framework for Similarity Search on the GPU - Technical Report
Venue: arXiv cs.DB 2018
Year: 2018

We propose a novel generic inverted index framework on the GPU (called GENIE), aiming to reduce the programming complexity of the GPU for parallel similarity search of different data types. Not every data type and similarity measure are supported by GENIE, but many popular ones are. We present the system design of GENIE, and demonstrate similarity search with GENIE on several data types along with a theoretical analysis of search results. A new concept of locality sensitive hashing (LSH) named $τ$-ANN search, and a novel data structure c-PQ on the GPU are also proposed for achieving this purpose. Extensive experiments on different real-life datasets demonstrate the efficiency and effectiveness of our framework. The implemented system has been released as open source.
Paired Threshold Graphs
Venue: arXiv cs.SI 2018
Year: 2018

Threshold graphs are recursive deterministic network models that have been proposed for describing certain economic and social interactions. One drawback of this graph family is that it has limited generative attachment rules. To mitigate this problem, we introduce a new class of graphs termed Paired Threshold (PT) graphs described through vertex weights that govern the existence of edges via two inequalities. One inequality imposes the constraint that the sum of weights of adjacent vertices has to exceed a specified threshold. The second inequality ensures that adjacent vertices have a weight difference upper bounded by another threshold. We provide a conceptually simple characterization and decomposition of PT graphs, analyze their forbidden induced subgraphs and present a method for performing vertex weight assignments on PT graphs that satisfy the defining constraints. Furthermore, we describe a polynomial-time algorithm for recognizing PT graphs. We conclude our exposition with an analysis of the intersection number, diameter and clustering coefficient of PT graphs.
Mixing Time for Some Adjacent Transposition Markov Chains
Venue: arXiv cs.DS 2018
Year: 2018

We prove rapid mixing for certain Markov chains on the set $S_n$ of permutations on $1,2,\dots,n$ in which adjacent transpositions are made with probabilities that depend on the items being transposed. Typically, when in state $σ$, a position $i<n$ is chosen uniformly at random, and $σ(i)$ and $σ(i{+}1)$ are swapped with probability depending on $σ(i)$ and $σ(i{+}1)$. The stationary distributions of such chains appear in various fields of theoretical computer science, and rapid mixing established in the uniform case. Recently, there has been progress in cases with biased stationary distributions, but there are wide classes of such chains whose mixing time is unknown. One case of particular interest is what we call the "gladiator chain," in which each number $g$ is assigned a "strength" $s_g$ and when $g$ and $g'$ are adjacent and chosen for possible swapping, $g$ comes out on top with probability $s_g/(s_g + s_{g'})$. We obtain a polynomial-time upper bound on mixing time when the gladiators fall into only three strength classes. A preliminary version of this paper appeared as "Mixing of Permutations by Biased Transposition" in STACS 2017.
Representing model inadequacy: A stochastic operator approach
Venue: arXiv cs.CE 2018
Year: 2018

Mathematical models of physical systems are subject to many uncertainties such as measurement errors and uncertain initial and boundary conditions. After accounting for these uncertainties, it is often revealed that discrepancies between the model output and the observations remain; if so, the model is said to be inadequate. In practice, the inadequate model may be the best that is available or tractable, and so despite its inadequacy the model may be used to make predictions of unobserved quantities. In this case, a representation of the inadequacy is necessary, so the impact of the observed discrepancy can be determined. We investigate this problem in the context of chemical kinetics and propose a new technique to account for model inadequacy that is both probabilistic and physically meaningful. A stochastic inadequacy operator $\mathcal{S}$ is introduced which is embedded in the ODEs describing the evolution of chemical species concentrations and which respects certain physical constraints such as conservation laws. The parameters of $\mathcal{S}$ are governed by probability distributions, which in turn are characterized by a set of hyperparameters. The model parameters and hyperparameters are calibrated using high-dimensional hierarchical Bayesian inference. We apply the method to a typical problem in chemical kinetics---the reaction mechanism of hydrogen combustion.
Two-level lot-sizing with inventory bounds
Venue: arXiv cs.CC 2018
Year: 2018

We study a two-level uncapacitated lot-sizing problem with inventory bounds that occurs in a supply chain composed of a supplier and a retailer. The first level with the demands is the retailer level and the second one is the supplier level. The aim is to minimize the cost of the supply chain so as to satisfy the demands when the quantity of item that can be held in inventory at each period is limited. The inventory bounds can be imposed at the retailer level, at the supplier level or at both levels. We propose a polynomial dynamic programming algorithm to solve this problem when the inventory bounds are set on the retailer level. When the inventory bounds are set on the supplier level, we show that the problem is NP-hard. We give a pseudo-polynomial algorithm which solves this problem when there are inventory bounds on both levels. In the case where demand lot-splitting is not allowed, i.e. each demand has to be satisfied by a single order, we prove that the uncapacitated lot-sizing problem with inventory bounds is strongly NP-hard. This implies that the two-level lot-sizing problems with inventory bounds are also strongly NP-hard when demand lot-splitting is considered.